a var investigation of currency composition in
TRANSCRIPT
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International Research Journal of Finance and Economics
ISSN 1450-2887 Issue 21 (2008)
EuroJournals Publishing, Inc. 2008
http://www.eurojournals.com/finance.htm
A VaR Investigation of Currency Composition in
Foreign Exchange Reserves
Jer-Shiou Chiou
Department of Finance and Banking, Shih-Chien University, Taipei, Taiwan
E-mail: [email protected]
Tel: 886-2-25381111 ext. 8927
Jui-Cheng Hung
Department of Finance and Banking
Yuanpei Institute of Science and Technology, Taiwan
E-mail: [email protected]
Mei-Maun Hseu
Department of Finance, Chihlee Institute of Techonology, Taipei,Taiwan
E-mail: [email protected]
Tel: 886-2--2253-7240 ext. 347
Abstract
In this study, Exponential Weighted Moving Average (EWMA), Bootstrapping, and
Monte Carlo Simulation are used to calculate the VaRs for three groups foreign reserves
portfolio from the year of 1995 to 2001.
Empirically we find that EWMA finished relatively better than the other two. Basedon developing countries currency in 2001 if the other currencies holding ratio is fixed,
reducing the U.S. dollar holding ratio while increasing the Euro holding ratio will makeVaR decrease in EWMA. However, if the Euro holding ratio is high, VaR increases. This
implies that despite the hedging effect of Euro, as its holding ratio increases, the marginal
effect decreases. The hedging effect of Euro is not persistent.
In general, increase the holdings of the lowest-valued component VaR currencywhile decreases the holdings of the highest-valued component VaR currency can in fact
reduce the risk of the portfolio.
Keywords: VaR, RMSE, RAPM, Marginal VaR, Component VaR
1. IntroductionExperiencing from the Asian financial crisis of 1997-98, most developing Asian economies haverapidly increased their external surpluses and accumulated foreign reserves to reduce their vulnerability
to future shocks; the expansion in US current account deficit is among the most. However, the
countries with largest reserves holdings were least affected by speculative pressures. But the recent
increasing oil prices and the foreign direct investment and portfolio inflows have begun to fade inseveral Asian countries, Asian governments suffer a sharp fall in the value of their dollar holdings and,more importantly, would see their export-dependent economies hit hard by a US slowdown. In this
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study, we seek to examine an alternative reserves management strategy in the level of internationalportfolio.
There are three reasons make central banks keep certain amount of foreign exchanges. Those
are liquidity needs for balance of trade, steady needs for government financing and interfering needsfor economy stability. Foreign exchanges have not only become a countrys key reserves assets but
also protect the countrys monetary interests. When the authority alters its portfolio holdings, the
context of safety, liquidity, and profitability in foreign exchange reserves have always been taken into
the consideration. Due to the most international business and liability takes U.S. dollars as majormeans; the U.S. dollar inevitably acts an essential part in a countrys foreign exchange reserves.
According to the IMFs annual report, from 1995 to 2001, there is over 60% of developing countries
foreign exchange reserves were U.S. dollars.On April 29, 1999, the World Bank Annual Conference, Pam Greens suggested Value-at-Risk
should be adopted when a countrys foreign reserves policies are under investigation. However, a
countrys actual reserves are usually confidential, not only retrieving truthful data becomes difficult,but the relative literatures are rare. Because the data retrieval difficulty, most studies focus on the
determinants in foreign reserves management. Beschloss and Mendes (1999) considered liquidity as
the main concern for central banks in foreign reserves management. Dooley, Lizondo, and Mathieson(1989) found the fixed exchange policy, the major international competitors, and the foreign debts are
the key determinants in reserves management decision. Barry and Donald (2000) had shown that thestability of foreign reserves was resulted from the commodity trading, the capital flow, and the fixedexchange policy. While Kenen (2002) suggested that as Euro increases their influential on the
members in the European Union; Euro still can hardly replace the U.S. dollar as the major international
currency.
In terms of risk management, Winfried (1999) decomposes the portfolio total risks intoseparated Value-at-Risks in parts, while Jos, Carlos, and Juan (2001) take the conceptual Value-at-
Risk in risk management practice. The study on center bank foreign reserves risk management only
appears in Blejer and Schumacher (1998) who theoretically construct a center banks investmentportfolio VaR evaluation model and then analyze the associates policies implications, but unfortunately
no empirical examinations were made.
Departure from the existing literatures, in this study we take nine1
regularly-taken reservescurrencies daily data, and the associates three major groups (that the IMF 2002 annual report
categorized as the whole world, the industrial countries and the developing countries) reserves
weighted information to study a central banks risk management strategy in terms of foreign reserves
portfolio. We adopt models of the Exponential Weighted Moving Average (EWMA), Bootstrapping,and Monte Carlo Simulation to compute the VaR for these three groups foreign reserves portfolio
from the year 1995 to 2001. Then, we take the best model to calculate and compare the risk-adjusted
performance measurement index (RAPM) for each group. At last, by adjusting each currenciesweighted importance, we analyze if the increasing Euro holdings was able to reduce the portfolios
risks.
2. Methodology2.1. Value at Risk
In measuring variations, variance and standard error have often been taken to reveal the magnitude in
the changes of future asset prices. As the fluctuation in future prices is inherent, potential gains andlosses in assets holding become inevitable. But most investors seem concern losses more than gains;
the variation criteria stated above is undesirable in describing this downside risks phenomenon. VaR is
defined as the worst expected loss over a great horizon within a given confidence level, Jorion (1996).
1 In practice, the IMFs annual report presents 8 currencies and 1 unspecified currency. In this study we take the Australia dollar as the unspecified
currency.
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It not only provides an aggregate statistic of the order of magnitude of potential losses due to marketrisk, but also summarizes the effects of leverage, diversification, and probabilities of adverse price
movements in a single dollar amount.
Lets define W0 as the initial investment at the beginning, and R*
represents the estimateexpected returns, then VaR can be written as:
meanVaR = ( )*0- W R - . (3)
If defined VaR as the absolute loss that excludes the expected returns, then VaR can berewritten as:
zeroVaR =*
0- W R . (4)
If we transform the probability distribution (W)f into a standard normal distribution (),
where ~ N(0, 1). Then the probability of possible returns which is less than W*
will be 1-C, and it can
be rewritten as:
d)(dr)r(fdw)w(fC1RW **
=== , (5)
wheret
Rt
*
= and t is the time factor. After the value of is determined, Value at Risk
can be written as: ( )ttWVaR 0zero = (6)tWVaR 0mean = . (7)
In this study we take zeroVaR as the measurement. That is, we take VaR as the absolute loss without
considering the expected returns.
2.2. EWMA with Variance-Covariance consideration
It had been shown in Jorion (2000) that when the returns are normally distributed, Value-at-Risk canbe expressed in two different forms depending on whether absolute returns or average returns were
taken. If foreign positions had been taken, the VaRs can be rewritten as
( )tttzero,1t ZWVaR = + (8)
ttmean,1t ZWVaR = + , (9)
where Wt is the assets value at time t, t is the average returns at time t, t is the standard deviation at
time t, and Z represents the critical value with a confidence level of 1-.
Under the definition of absolute return and the assumption of W t=1, Z, t and t can then beobtained. If a parametric model was taken to compute the VaR for a single asset, the variances multiple
by a critical value at the confidence level of 1- is necessary. While in computing the VaR for certain
portfolios, the influential of covariance has to be considered, other than the variance.We therefore take EWMA as a mean to estimate the fluctuations of the portfolio. Their variance
and covariance can thus be rewritten as:2
1t,i
2
1t,i
2
t,i r)1( += (10)
1t,j1t,i1t,ijt,ij rr)1( += , (11)
where ji , 2,ti and tij, are the variance of asset i and the covariance of asset i and j at time t. 1, tir
and 1, tjr are the return ratios of asset i and j at time t-1. is the decay factor of which 0.94 for the
daily data and 0.97 for the monthly data. When EWMA is in use, the VaR for the next T days can bewritten as:
TZWVaR t,ptt = , (12)
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where = wwtP ',
NN
2
t,Nt,2Nt,1N
t,N2
2
t,2t,21
t,N1t,12
2
t,1
=L
MOMM
L
K
1NN
2
1
w
w
w
w
=M
;
iw is the weighted ratio of the ith
foreign asset to the total portfolio, i,t, j,t, and ij,t are obtained by
EWMA and they are the variance and covariance of the foreign asset, respectively. tP, is the standarddeviation for the portfolio at time t, while T is the holding period for the foreign asset. Under the
assumption of the holding period for the foreign asset is 1 day (T=1) and the confidence level is 99%,
the daily VaR is obtained.
2.3. Bootstrapping
It is not necessary to fully understand the distribution of the population when Bootstrapping is in usedto calculate the underlying VaR. The method takes the limited historical returns and go through the
iterate sampling to construct the asset portfolios distribution for future returns. The VaR of an asset
portfolio is then obtainable, whenever the confidence level and holding period are specified. The
difference between Bootstrapping and Historical Simulation is that Historical Simulation directlyutilizes the future returns distribution that has only one price path, while Bootstrapping makes an
iterate sampling from historical data to simulate the real returns distribution to improve the
shortcoming of a Historical Simulation.Assume it is known for each assets prices in a portfolio for the last 251 days. It would be
attainable to get next days VaR for the portfolio by the Bootstrapping. The procedure can be described
as follows:
Step 1
Convert the historical 251 days prices into 250 returns and perform 10000 repeated sampling from the
underlying returns. The same process can apply to any portfolio which contains N assets.
)T(Ri , where i 1, 2...,N= T 1,2,...,10000=
Step 2Multiply the returns by the corresponding importance ratio, which is measured by any single asset to
the portfolio. Sorting the data in increasing order and sum together, the distribution of next days return
can therefore be obtained.
Step 3
Using the above simulated future returns, VaR can be computed by a percentile criterion on the
confidence level of 1-.
2.4. Monte Carlo Simulation
Before applying the Monte Carlo Simulation, it is necessary to assume that the underlying portfoliosreturn follows a certain random process. Usually, the Geometric Brownian Motion Model (GBM) is
applicable to the assets such as stocks and foreign exchange. It can be expressed as:
dZdtR
dRtt
t
t += , (13)
where Rt is the return of portfolio, t is the drift term of the portfolio at time t, t is the standard
deviation at time t, and dZ is normally distributed with 0 mean and dt variance, whereby ( )dt,0N~dZ .From (13) we found that if a portfolios return follows a multivariable normal distribution that
is ( )NN1N ,N~R , then in order to simulate the return as an N N normal distribution, a MonteCarlo process as be described as follows:
Step 1
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Take the Cholesky Decomposition to decompose an NN variance- covariance matrix as =TAAlike follows
=
NN2N1N
2221
11
aaa
0
aa
00a
A
L
OMM
ML
L
=
NN
2N22
1N2111
T
a00
aa0
aaa
A
L
MOMM
L
L
=
2
N2N1N
N2
2
221
N112
2
1
L
MOMM
L
L
.
Step 2
An 1N Z matrix can be generated, where Z is a multivariate standard normal distributed randomvariable, and ( )NI,0N~Z , where
=
100
00
001
IN O .
Step 3
Multiplying the A matrix from step 1 by the Z matrix, we can get R=AZ, which is an N multivariablenormal random variable. Here the covariance matrix of R is
( ) ( ) ( ) ( ) TTTTT AZZAEAAZZERRERVAR === = == TTN AAAAI .Step 4
Repeat step (3) 10000 times and get a sample size of 10000 next days portfolios return. Sorting the
data in increasing order, we get the distribution for the portfolios next days return. VaR can thereforebe computed by a percentile criterion on the confidence level of 1- for this simulated future returns.
2.5. VaR, Evaluation and Performance
In VaRs evaluation and its performance effectiveness, we adopt Kupiecs (1995) Proportion of Failure
Test (PF-Test). This gives the verification of whether the constructed 0 was consistent with the actual
. Here, the null hypothesis is H0 0 = , while the statistics are
)])1((ln))1((ln[2LR xn0x
0
xnx
PF
= ~ )1(2 , (14)
where 0 is the exception rate, n is the number of observations, x is the counts which shown the actual
return greater than the computed VaR ratio in the models, and nx = is the ratio of which actualreturns are greater than the VaR.
After the underlying VaR models passed the PF-Test, the prediction effectiveness for the
models is investigated then. Forward-testing and the efficiency of funds uses are taken to be the
measurement indices.Next, we take the Root Mean Square Error (RMSE) as the criterion to determine the usage
efficiency of funds in the short run. This can be seemed as the square root of the average sums of
square of differences between the predicted value and actual value. This implies that when we acceptthe condition of a reliable VaR model, the smaller RMSE is the closer is a VaR from the actual loss. It
also shows that there are no excess reserve funds to compensate for the possible loss in the short run.
For a reliable VaR model with a smaller RMSE, this implies that there are certain advantages on bothrisk control and funds usage. RMSE can be written as
( )
n
VaRr
RMSE
n
1t
2
tt=
= , (15)
where rt is the actual returns and VaRt is the value computed from the model.
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Benet (1992) investigated foreign exchange futures, and suggested that if out-of-sample or ex-
ante were taken to evaluate the hedging effect, external effect should be emphasized in order to be
more meaningful for the investors. Similarly, forward-testing is based on ex-ante and consider thefuture potential losses.
2.6. The Decomposition of VaR
The main reason for forming a portfolio is hoping through investing in different assets investor candiversify the markets non-systematic risk and minimize the investment risk, whereby VaR can pre-
reveal the possible loss for the portfolio. Since VaR can integrate all of the possible holding portionsfor an institution and monitor the volume of risks exposure. And makes the risk of the investment
position fulfills the requirement for the limits of the capital possible. VaR decomposition hence reveals
the importance of any single asset in an investment portfolio. Those include the individual VaR, the
marginal VaR, incremental VaR, and the component VaR. A brief discussion is as follows.
2.6.1. Individual VaR
Assume that there are N assets within a portfolio. Any single assets VaR can be written as
iii ZwVaR = ,
where N,,1i K= , 1wN
1i
i ==
. The total risk for the portfolio is =
=N
1i
iP VaRVaR , without considering
the interaction among the assets. But in fact, every coefficient between any two assets should lie
between 1 and 1. Therefore, theoretically total risk for the portfolio should less than the sum of every
assets VaR, that can be written as =